We conclude this subsection with a result that may be of independent interest. Theorems REF  and REF  imply that a particular optimality condition for minimizing separable convex functions over (integral) base polyhedra is unique for these problems. For a given feasible set \(C\)  and a vector \(x \in C\) , this condition is as follows (see also, e.g., Theorem 8.1 in [1]})
